Geodesic
Fast Basins and Branched Fractal Manifolds of Attractors of Iterated Function Systems
Symmetry, integrability and geometry: methods and applications, Tome 11 (2015) Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

The fast basin of an attractor of an iterated function system (IFS) is the set of points in the domain of the IFS whose orbits under the associated semigroup intersect the attractor. Fast basins can have non-integer dimension and comprise a class of deterministic fractal sets. The relationship between the basin and the fast basin of a point-fibred attractor is analyzed. To better understand the topology and geometry of fast basins, and because of analogies with analytic continuation, branched fractal manifolds are introduced. A branched fractal manifold is a metric space constructed from the extended code space of a point-fibred attractor, by identifying some addresses. Typically, a branched fractal manifold is a union of a nondenumerable collection of nonhomeomorphic objects, isometric copies of generalized fractal blowups of the attractor.
Keywords: iterated function system; fast basins; fractal continuation; fractal manifold. 
@article{SIGMA_2015_11_a83,
     author = {Michael F. Barnsley and Andrew Vince},
     title = {Fast {Basins} and {Branched} {Fractal} {Manifolds} of {Attractors} of {Iterated} {Function} {Systems}},
     journal = {Symmetry, integrability and geometry: methods and applications},
     year = {2015},
     volume = {11},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SIGMA_2015_11_a83/}
}
TY  - JOUR
AU  - Michael F. Barnsley
AU  - Andrew Vince
TI  - Fast Basins and Branched Fractal Manifolds of Attractors of Iterated Function Systems
JO  - Symmetry, integrability and geometry: methods and applications
PY  - 2015
VL  - 11
UR  - http://geodesic.mathdoc.fr/item/SIGMA_2015_11_a83/
LA  - en
ID  - SIGMA_2015_11_a83
ER  - 
%0 Journal Article
%A Michael F. Barnsley
%A Andrew Vince
%T Fast Basins and Branched Fractal Manifolds of Attractors of Iterated Function Systems
%J Symmetry, integrability and geometry: methods and applications
%D 2015
%V 11
%U http://geodesic.mathdoc.fr/item/SIGMA_2015_11_a83/
%G en
%F SIGMA_2015_11_a83
Michael F. Barnsley; Andrew Vince. Fast Basins and Branched Fractal Manifolds of Attractors of Iterated Function Systems. Symmetry, integrability and geometry: methods and applications, Tome 11 (2015). http://geodesic.mathdoc.fr/item/SIGMA_2015_11_a83/

[1] Barnsley M. F., “Transformations between self-referential sets”, Amer. Math. Monthly, 116 (2009), 291–304, arXiv: math.DS/0703398 | DOI | MR | Zbl

[2] Barnsley M. F., Leśniak K., Basic topological structure of fast basins, arXiv: 1308.4230

[3] Barnsley M. F., Leśniak K., Vince A., Symbolic iterated function systems, fast basins and fractal manifolds, arXiv: 1308.3819v1

[4] Barnsley M. F., Vince A., “The chaos game on a general iterated function system”, Ergodic Theory Dynam. Systems, 31 (2011), 1073–1079, arXiv: 1005.0322 | DOI | MR | Zbl

[5] Barnsley M. F., Vince A., “Developments in fractal geometry”, Bull. Math. Sci., 3 (2013), 299–348 | DOI | MR | Zbl

[6] Barnsley M. F., Vince A., “Fractal continuation”, Constr. Approx., 38 (2013), 311–337, arXiv: 1209.6100 | DOI | MR | Zbl

[7] Barnsley M. F., Vince A., “The Conley attractors of an iterated function system”, Bull. Aust. Math. Soc., 88 (2013), 267–279, arXiv: 1206.6319 | DOI | MR | Zbl

[8] Barnsley M. F., Vince A., “Fractal tilings from iterated function systems”, Discrete Comput. Geom., 51 (2014), 729–752, arXiv: 1310.6344 | DOI | MR | Zbl

[9] Hata M., “On the structure of self-similar sets”, Japan J. Appl. Math., 2 (1985), 381–414 | DOI | MR | Zbl

[10] Hutchinson J. E., “Fractals and self-similarity”, Indiana Univ. Math. J., 30 (1981), 713–747 | DOI | MR | Zbl

[11] Ionescu M., Kumjian A., “Groupoid actions on fractafolds”, SIGMA, 10 (2014), 068, 14 pp., arXiv: 1311.3880 | DOI | MR | Zbl

[12] Kieninger B., Iterated function systems on compact Hausdorff spaces, Ph.D. Thesis, Augsburg University, Germany, 2002 | Zbl

[13] Mandelbrot B. B., The fractal geometry of nature, W.H. Freeman and Co., San Francisco, Calif., 1982 | MR | Zbl

[14] Strichartz R. S., “Fractals in the large”, Canad. J. Math., 50 (1998), 638–657 | DOI | MR | Zbl

[15] Strichartz R. S., “Fractafolds based on the Sierpiński gasket and their spectra”, Trans. Amer. Math. Soc., 355 (2003), 4019–4043 | DOI | MR | Zbl

[16] Strichartz R. S., Differential equations on fractals, Princeton University Press, Princeton, NJ, 2006 | MR | Zbl

[17] Vince A., “Möbius iterated function systems”, Trans. Amer. Math. Soc., 365 (2013), 491–509 | DOI | MR | Zbl